In addition to these well-posedness results, we also obtain a Hopf-Lax variational representation for the solution to the infinite-dimensional Hamilton-Jacobi equation (REF ) in the case when the non-linearity \(\infty \)  is convex. Hopf-Lax formulas for related problems have been explored in [1]}, [2]}, [3]}, [4]}. In [5]}, this variational representation will allow us to verify that, in the disassortative regime, our conjectured asymptotic mutual information for the sparse stochastic block model coincides with the value of the asymptotic mutual information established in [6]}. The convexity condition on \(\infty \)  boils down to the requirement that the mapping \((x,y) \mapsto g(xy)\)  be non-negative definite, and can be phrased as follows.
